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Working bi-Hamiltonian structure and Jacobi identity in Frenet-Serret frameassociated to a dynamical system, we proved that all dynamical systems in threedimensions possess two compatible Poisson structures. We investigate relationsbetween geometry of surfaces defined by potential function of a gradient systemand its bi-Hamiltonian structure. We show that it is possible to findHamiltonian functions whose gradient flows have geodesic curvature zero onpotential surfaces. Using this, we conclude that Hamiltonian functions aredetermined by distance functions on potential surfaces. We apply this techniqueto find conserved quantities of three dimensional gradient systems includingthe Aristotelian model of the three-body motion.

Bi-Hamiltonian Structure of Gradient Systems in Three Dimensions and Geometry of Potential Surfaces